# Properties

 Label 1323.2.a.g Level $1323$ Weight $2$ Character orbit 1323.a Self dual yes Analytic conductor $10.564$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + 4q^{5} + 3q^{8} + O(q^{10})$$ $$q - q^{2} - q^{4} + 4q^{5} + 3q^{8} - 4q^{10} - 2q^{11} + q^{13} - q^{16} + 6q^{17} + 4q^{19} - 4q^{20} + 2q^{22} - 6q^{23} + 11q^{25} - q^{26} - 2q^{29} + 3q^{31} - 5q^{32} - 6q^{34} + 3q^{37} - 4q^{38} + 12q^{40} + 2q^{41} - q^{43} + 2q^{44} + 6q^{46} - 6q^{47} - 11q^{50} - q^{52} + 6q^{53} - 8q^{55} + 2q^{58} - 6q^{59} - 5q^{61} - 3q^{62} + 7q^{64} + 4q^{65} + 7q^{67} - 6q^{68} - 6q^{73} - 3q^{74} - 4q^{76} + 11q^{79} - 4q^{80} - 2q^{82} - 6q^{83} + 24q^{85} + q^{86} - 6q^{88} + 4q^{89} + 6q^{92} + 6q^{94} + 16q^{95} + 9q^{97} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 0 −1.00000 4.00000 0 0 3.00000 0 −4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.a.g 1
3.b odd 2 1 1323.2.a.m 1
7.b odd 2 1 1323.2.a.d 1
7.c even 3 2 189.2.e.c yes 2
21.c even 2 1 1323.2.a.p 1
21.h odd 6 2 189.2.e.a 2
63.g even 3 2 567.2.g.e 2
63.h even 3 2 567.2.h.b 2
63.j odd 6 2 567.2.h.e 2
63.n odd 6 2 567.2.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.a 2 21.h odd 6 2
189.2.e.c yes 2 7.c even 3 2
567.2.g.b 2 63.n odd 6 2
567.2.g.e 2 63.g even 3 2
567.2.h.b 2 63.h even 3 2
567.2.h.e 2 63.j odd 6 2
1323.2.a.d 1 7.b odd 2 1
1323.2.a.g 1 1.a even 1 1 trivial
1323.2.a.m 1 3.b odd 2 1
1323.2.a.p 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2} + 1$$ $$T_{5} - 4$$ $$T_{13} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$T$$
$11$ $$2 + T$$
$13$ $$-1 + T$$
$17$ $$-6 + T$$
$19$ $$-4 + T$$
$23$ $$6 + T$$
$29$ $$2 + T$$
$31$ $$-3 + T$$
$37$ $$-3 + T$$
$41$ $$-2 + T$$
$43$ $$1 + T$$
$47$ $$6 + T$$
$53$ $$-6 + T$$
$59$ $$6 + T$$
$61$ $$5 + T$$
$67$ $$-7 + T$$
$71$ $$T$$
$73$ $$6 + T$$
$79$ $$-11 + T$$
$83$ $$6 + T$$
$89$ $$-4 + T$$
$97$ $$-9 + T$$